Feng Bao

Zezhong Zhang, Feng Bao, Guannan Zhang

The Ensemble Score Filter (EnSF) is a score-based diffusion model approach for solving high-dimensional and nonlinear data assimilation problems. While initial applications of EnSF to the Lorenz-96 model and the quasi-geostrophic system showed potential, the current method employs a heuristic weighted sum to combine the prior and the likelihood score functions. This introduces a structural error into the estimation of the posterior score function in the nonlinear setting. This work addresses this challenge by developing an iterative ensemble score filter (IEnSF) that applies an iterative algorithm as an outer loop around the reverse-time stochastic differential equation solver. When the state dynamics or the observation operator is nonlinear, the iterative algorithm can gradually reduce the posterior score estimation error by improving the accuracy of approximating the conditional expectation of the likelihood score function. The number of iterations required depends on the distance between the prior and posterior distributions. Numerical experiments demonstrate that the IEnSF algorithm substantially reduces the error in posterior score estimation in the nonlinear setting and thus improves the accuracy of tracking high-dimensional dynamical systems.

Richard Archibald, Feng Bao, Peter Maksymovych

The connection between forward backward doubly stochastic differential equations and the optimal filtering problem is established without using the Zakai's equation. The solutions of forward backward doubly stochastic differential equations are expressed in terms of conditional law of a partially observed Markov diffusion process. It then follows that the adjoint time-inverse forward backward doubly stochastic differential equations governs the evolution of the unnormalized filtering density in the optimal filtering problem.

Ryan Bausback, Jingqiao Tang, Lu Lu et al.

We develop a novel framework for uncertainty quantification in operator learning, the Stochastic Operator Network (SON). SON combines the stochastic optimal control concepts of the Stochastic Neural Network (SNN) with the DeepONet. By formulating the branch net as an SDE and backpropagating through the adjoint BSDE, we replace the gradient of the loss function with the gradient of the Hamiltonian from Stohastic Maximum Principle in the SGD update. This allows SON to learn the uncertainty present in operators through its diffusion parameters. We then demonstrate the effectiveness of SON when replicating several noisy operators in 2D and 3D.

Toan Huynh, Ruth Lopez Fajardo, Guannan Zhang et al.

We propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central role in modeling complex physical systems under uncertainty, but their numerical solutions often suffer from model errors and reduced accuracy due to incomplete physical knowledge and environmental variability. To address these challenges, we encode the governing physics into the score function of a diffusion model using simulation data and incorporate observational information via a likelihood-based correction in a reverse-time stochastic differential equation. This enables adaptive learning through iterative refinement of the solution as new data becomes available. To improve computational efficiency in high-dimensional settings, we introduce the ensemble score filter, a training-free approximation of the score function designed for real-time inference. Numerical experiments on benchmark SPDEs demonstrate the accuracy and robustness of the proposed method under sparse and noisy observations.

Jingqiao Tang, Ryan Bausback, Feng Bao et al.

Federated learning is a machine learning paradigm that leverages edge computing on client devices to optimize models while maintaining user privacy by ensuring that local data remains on the device. However, since all data is collected by clients, federated learning is susceptible to latent noise in local datasets. Factors such as limited measurement capabilities or human errors may introduce inaccuracies in client data. To address this challenge, we propose the use of a stochastic neural network as the local model within the federated learning framework. Stochastic neural networks not only facilitate the estimation of the true underlying states of the data but also enable the quantification of latent noise. We refer to our federated learning approach, which incorporates stochastic neural networks as local models, as Federated stochastic neural networks. We will present numerical experiments demonstrating the performance and effectiveness of our method, particularly in handling non-independent and identically distributed data.

Feng Bao, Zezhong Zhang, Guannan Zhang

We propose an ensemble score filter (EnSF) for solving high-dimensional nonlinear filtering problems with superior accuracy. A major drawback of existing filtering methods, e.g., particle filters or ensemble Kalman filters, is the low accuracy in handling high-dimensional and highly nonlinear problems. EnSF attacks this challenge by exploiting the score-based diffusion model, defined in a pseudo-temporal domain, to characterizing the evolution of the filtering density. EnSF stores the information of the recursively updated filtering density function in the score function, instead of storing the information in a set of finite Monte Carlo samples (used in particle filters and ensemble Kalman filters). Unlike existing diffusion models that train neural networks to approximate the score function, we develop a training-free score estimation that uses a mini-batch-based Monte Carlo estimator to directly approximate the score function at any pseudo-spatial-temporal location, which provides sufficient accuracy in solving high-dimensional nonlinear problems as well as saves a tremendous amount of time spent on training neural networks. High-dimensional Lorenz-96 systems are used to demonstrate the performance of our method. EnSF provides surprising performance, compared with the state-of-the-art Local Ensemble Transform Kalman Filter method, in reliably and efficiently tracking extremely high-dimensional Lorenz systems (up to 1,000,000 dimensions) with highly nonlinear observation processes.

Feng Bao, Yanzhao Cao, Clayton Webster et al.

In this paper, we propose a meshfree approximation method for the implicit filter developed in [2], which is a novel numerical algorithm for nonlinear filtering problems. The implicit filter approximates conditional distributions in the optimal filter over a deterministic state space grid and is developed from samples of the current state obtained by solving the state equation implicitly. The purpose of the meshfree approximation is to improve the efficiency of the implicit filter in moderately high-dimensional problems. The construction of the algorithm includes generation of random state space points and a meshfree interpolation method. Numerical experiments show the effectiveness and efficiency of our algorithm.